In a recent interview, Dragon Ball creator Akira Toriyama revealed that for Saiyans to become powerful Super Saiyans, they need to have enough S-Cells.
When asked if anyone who trains enough can become a Super Saiyan, Toriyama replied (via @Herms98), “It’s not like anyone can become a Super Saiyan through training and anger.
Thus, one of the reasons why Goku and Vegeta’s kids became Super Saiyans is that they inherited these cells.
“Most Saiyans have some S-Cells, although not a great quantity,” Toriyama continued (via @Herms98).